congruence

Definitions

nCongruenceSuitableness of one thing to another; agreement; consistency.

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Century Dictionary and Cyclopedia

ncongruenceSuitableness or appropriateness of one thing to another; agreement; consistency. Also congruency.

ncongruenceTn mathematics, a relation between three numbers such that the difference between two of them, which are said to be congruent, is divisible by the third, which is called the modulus. The following example shows the mode of writing a congruence: x − 1 ≡ (x − 1)(x − 2)(x − 3)(x − 4)(x − 5)(x − 6) (mod. 7), which means that any integer being substituted for x, the remainders of the quantities on the two sides of the sign ≡ after division by 7 are equal. See congruency.

ncongruenceIn grammar, concord; agreement.

ncongruenceSame as congruency, 2.

ncongruenceIn geometry, identity in shape and size. Its symbol is ≡.

ncongruenceIn line geometry, a set of ∞ lines, such that any two given conditions determine a definite finite number of lines of the set.

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Chambers's Twentieth Century Dictionary

nsCongruenceagreement: suitableness

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Etymology

Webster's Revised Unabridged Dictionary

L. congruentia,: cf. OF. cornguence,

Chambers's Twentieth Century Dictionary

L. congruĕre, to run together.

Usage

In literature:

To wryte orthographicallie ther are to be considered the symbol, the thing symbolized, and their congruence.

"Of the Orthographie and Congruitie of the Britan Tongue" by Alexander Hume

Congruence depends on motion, and thereby is generated the connexion between spatial congruence and temporal congruence.

"The Concept of Nature" by Alfred North Whitehead

Therefore of a congruence From hence thou must have my heart and obedience.

"Everyman and Other Old Religious Plays, with an Introduction" by Anonymous

And you, Charles, let me hope your feelings are in solemn congruence with this momentous step.

"The Works of Robert Louis Stevenson, Volume XV" by Robert Louis Stevenson

It follows that all lines in which corresponding planes in two projective pencils meet form a congruence.

In poetry:

might fairly achieve
it. As for butterflies, I can hardly conceive
of one's attending upon you, but to question
the congruence of the complement is vain, if it exists.

"To a Steam Roller" by Marianne Moore

In science:

We deﬁne abelian extensions of algebras in congruence-modular varieties.

Abelian extensions of algebras in congruence-modular varieties

We have shown that these groups are isomorphic for V congruence-modular, but will not give the proof in this paper as it is quite tedious.

Abelian extensions of algebras in congruence-modular varieties

The kernel congruence of a homomorphism f will be denoted by ker f .

Abelian extensions of algebras in congruence-modular varieties

We denote the greatest and least congruences of A by ⊤A and ⊥A, and the identity homomorphism of A by 1A .

Abelian extensions of algebras in congruence-modular varieties

The commutator of two congruences α, β ∈ Con A is denoted by [α, β ]. A congruence α is said to be abelian if [α, α] = ⊥A .

Abelian extensions of algebras in congruence-modular varieties

If V is a congruence-modular variety of algebras, a ternary term d is called a difference term for V if 1. d(x, x, y ) = y is an identity of V, and 2. for all A ∈ V, θ ∈ Con A, and x, y ∈ A such that x θ y, we have d(x, y, y ) [θ, θ] x.

Abelian extensions of algebras in congruence-modular varieties

At least one such term exists for any congruence-modular variety.

Abelian extensions of algebras in congruence-modular varieties

Theorem 1.1. () Let V be a congruence-modular variety of algebras, and A ∈ V.

Abelian extensions of algebras in congruence-modular varieties

Let γ : E → ˜E be a homomorphism, and let α and ˜α be congruences of E and ˜E, respectively, such that γ (α) ⊆ ˜α.

Abelian extensions of algebras in congruence-modular varieties

This is a congruence-modular variety, as can easily be proved, because of the underlying abelian group structures.

Abelian extensions of algebras in congruence-modular varieties

This was done in the generality of V an arbitrary (i.e., not necessarily congruence-modular) variety of algebras of some type.

Abelian extensions of algebras in congruence-modular varieties

With the last property included, the equivalence relation shares an additional well-known property with congruence relations.

Factorization of integers and arithmetic functions

According to the congruence-condition in the previous section, the product α1 ∗ β1 is in the same class with every other product of the form α2 ∗ β2 where α2 is any member of α, and β2 is any member of β .

Factorization of integers and arithmetic functions

Gauss proves it [13, II, 14, 15] using congruences, also based on division with remainder and diﬃcult to achieve without an order relation.

Factorization of integers and arithmetic functions

Or, for a congruence on an ob ject A, i. e. a parallel pair (a1, a2 ) : R ⇉ A such that the resulting map ha1 (−), a2 (−)i : hom(X, R) → hom(X, A) × hom(X, A) is an equivalence relation for any X, we might use notation xRy or x ∼R y, or just x ∼ y for morphisms x, y : X → A such that (x, y ) factors through R.